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Chapter 1: Q. 39 (page 135)

Calculate each of the limits in Exercises $29 - 70$ .
$\lim_{x 鈫� \frac{1}{2}} \frac{4 x - 2}{6 x^{2} + x - 2}$ .

Short Answer

Expert verified

The value of $\lim_{x 鈫� \frac{1}{2}} \frac{4 x - 2}{6 x^{2} + x - 2}$ is, $\frac{4}{7}$ .

Step by step solution

01

Step 1. Given information

$\lim_{x 鈫� \frac{1}{2}} \frac{4 x - 2}{6 x^{2} + x - 2}$ .

02

Step 2. The limit is calculated as below:

$\lim_{x 鈫� \frac{1}{2}} \frac{4 x - 2}{6 x^{2} + x - 2} = \lim_{x 鈫� 1 / 2} \frac{4 x - 2}{6 x^{2} + 4 x - 3 x - 2} = \lim_{x 鈫� 1 / 2} \frac{2 (2 x - 1)}{2 x (3 x + 2) - 1 (3 x + 2)} = \lim_{x 鈫� 1 / 2} \frac{2 (2 x - 1)}{(3 x + 2) (2 x - 1)} = \lim_{x 鈫� 1 / 2} \frac{2}{(3 x + 2)}$

03

Step 3. Let us use the quotient rule in the obtained equation.

$\lim_{x 鈫� 1 / 2} \frac{2}{(3 x + 2)} = \frac{\lim_{x 鈫� 1 / 2} (2)}{\lim_{x 鈫� 1 / 2} (3 x + 2)} = \frac{2}{3 (\frac{1}{2}) + 2} = \frac{2}{\frac{3}{2} + 2} = \frac{2}{\frac{7}{2}} = \frac{2 (2)}{7} = \frac{4}{7}$

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Most popular questions from this chapter

Calculate each of the limits:

$\lim_{x 鈫� 3} \frac{\sqrt{x}}{\tan^{- 1} \sqrt{x}}$

Write delta-epsilon proofs for each of the limit statements $\lim_{x 鈫� c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x 鈫� 1} (2 x^{2} - 4 x + 3) = 1$ .

Calculate each of limits:

$\lim_{x 鈫� 1} \frac{1}{\sin^{- 1} x}$ .

For each limit statement in Exercises $41 - 44$ , use algebra to find $未$ or $N$ in terms of $蔚$ or $M$ , according to the appropriate formal limit definition.

$\lim_{x 鈫� 鈭�} (x^{2} + 2) = 鈭�$ ,find $N$ in terms of $M$ .

Use algebra to find the largest possible value of 未 or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.
$If x > N, then 1 鈭� 2 x < 鈭� 500$

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